xaddge0 - Intuitionistic Logic Explorer

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Theorem xaddge0 10259

Description: The sum of nonnegative extended reals is nonnegative. (Contributed by Mario Carneiro, 21-Aug-2015.)

**Assertion**
Ref	Expression
xaddge0	⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 +_𝑒 𝐵))

**Proof of Theorem xaddge0**
Step	Hyp	Ref	Expression
1		0xr 8362	. . 3 ⊢ 0 ∈ ℝ^*
2	1	a1i 9	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ∈ ℝ^)
3		simplr 533	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ^)
4		xaddcl 10241	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
5	4	adantr 276	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
6		simprr 537	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐵)
7		xaddid2 10244	. . . 4 ⊢ (𝐵 ∈ ℝ^* → (0 +_𝑒 𝐵) = 𝐵)
8	3, 7	syl 14	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → (0 +_𝑒 𝐵) = 𝐵)
9		simpll 531	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐴 ∈ ℝ^)
10		simprl 535	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐴)
11		xleadd1a 10254	. . . 4 ⊢ (((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 0 ≤ 𝐴) → (0 +_𝑒 𝐵) ≤ (𝐴 +_𝑒 𝐵))
12	2, 9, 3, 10, 11	syl31anc 1281	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → (0 +_𝑒 𝐵) ≤ (𝐴 +_𝑒 𝐵))
13	8, 12	eqbrtrrd 4149	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ≤ (𝐴 +_𝑒 𝐵))
14	2, 3, 5, 6, 13	xrletrd 10193	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 +_𝑒 𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1402 ∈ wcel 2209 class class class wbr 4125 (class class class)co 6075 0cc0 8169 ℝ^*cxr 8349 ≤ cle 8351 +_𝑒 cxad 10151

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-addcom 8269 ax-addass 8271 ax-i2m1 8274 ax-0id 8277 ax-rnegex 8278 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-id 4433 df-po 4436 df-iso 4437 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-fv 5380 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-xadd 10154

This theorem is referenced by: ge0xaddcl 10364

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